Why does $i$ appear in this solution?

Question

Solving the equation

$x^s=(1-x)^s$ yields:

This supposedly gives the locations for the intersections between $x^s$ and $(1-x)^s.$ Why does $i$ appear in this solution?

Answer 1

It appears because they consider all the complex solutions.

$$x^s = (1-x)^s$$

$$\left(\frac{1-x}{x} \right)^s = 1$$

Now recall that $y^s=1$ has $s$ distinct complex roots of unity which are of the form of $\exp\left( \frac{2k\pi j}{s}\right)$ where $j \in \{0, \ldots, s-1\}$.

What do you do if you just want a real solution? Consider the cases where $s$ is even or odd.

If $s$ is even, we have $\frac{1-x}{x} \in \{ 1, -1\}$ and you can rule out one of them.

If $s$ is odd, we have $\frac{1-x}{x} = 1$.